| L(s) = 1 | − 1.78·5-s + 0.353·7-s − 5.54·13-s − 3.81·17-s + 4.61·19-s + 7.00·23-s − 1.82·25-s − 2.11·29-s + 3.45·31-s − 0.629·35-s − 1.74·37-s + 1.48·41-s − 3.92·43-s + 1.21·47-s − 6.87·49-s − 7.92·53-s + 3.09·59-s − 7.60·61-s + 9.88·65-s + 1.17·67-s + 2.36·71-s + 3.90·73-s + 4.10·79-s − 0.818·83-s + 6.79·85-s + 1.92·89-s − 1.95·91-s + ⋯ |
| L(s) = 1 | − 0.796·5-s + 0.133·7-s − 1.53·13-s − 0.924·17-s + 1.05·19-s + 1.46·23-s − 0.365·25-s − 0.393·29-s + 0.620·31-s − 0.106·35-s − 0.286·37-s + 0.231·41-s − 0.599·43-s + 0.176·47-s − 0.982·49-s − 1.08·53-s + 0.402·59-s − 0.973·61-s + 1.22·65-s + 0.143·67-s + 0.281·71-s + 0.456·73-s + 0.462·79-s − 0.0898·83-s + 0.736·85-s + 0.204·89-s − 0.205·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140229904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140229904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 - 0.353T + 7T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 - 7.00T + 23T^{2} \) |
| 29 | \( 1 + 2.11T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 1.74T + 37T^{2} \) |
| 41 | \( 1 - 1.48T + 41T^{2} \) |
| 43 | \( 1 + 3.92T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 - 3.09T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 - 4.10T + 79T^{2} \) |
| 83 | \( 1 + 0.818T + 83T^{2} \) |
| 89 | \( 1 - 1.92T + 89T^{2} \) |
| 97 | \( 1 + 9.47T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77834861489071193302926493930, −7.09237985900176607541146376214, −6.65103764887320010028917581826, −5.53808130275182971048731869827, −4.87232676689063670248315897461, −4.40498031190517367672321181000, −3.38688161702604043106212705225, −2.76629977195572236777131322068, −1.76870646721586743051554253361, −0.50825237092577413999404718299,
0.50825237092577413999404718299, 1.76870646721586743051554253361, 2.76629977195572236777131322068, 3.38688161702604043106212705225, 4.40498031190517367672321181000, 4.87232676689063670248315897461, 5.53808130275182971048731869827, 6.65103764887320010028917581826, 7.09237985900176607541146376214, 7.77834861489071193302926493930
